Question: All of the 4th grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$3.00$ each for students, and the group paid $$40.50$ in total. The next month, the same group visited a science museum where the tickets cost $$22.00$ each for teachers and $$7.50$ each for students, and the group paid $$126.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+3y = 40.5}$ ${22x+7.5y = 126}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-22x-12y = -162}$ ${22x+7.5y = 126}$ Add the top and bottom equations together. $ -4.5y = -36 $ $ y = \dfrac{-36}{-4.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {5.5x+3y = 40.5}$ to find $x$ ${5.5x + 3}{(8)}{= 40.5}$ $5.5x+24 = 40.5$ $5.5x = 16.5$ $x = \dfrac{16.5}{5.5}$ ${x = 3}$ You can also plug ${y = 8}$ into $ {22x+7.5y = 126}$ and get the same answer for $x$ ${22x + 7.5}{(8)}{= 126}$ ${x = 3}$ There were $3$ teachers and $8$ students on the field trips.